on the performance of coupled and segregated …€¦ · scv1 scv2 scv3 scv4 figure 1: control...

Proceedings of the ENCIT 2014Copyright c© 2014 by ABCM

15th Brazilian Congress of Thermal Sciences and EngineeringNovember 10-13, 2014, Belém, PA, Brazil

ON THE PERFORMANCE OF COUPLED AND SEGREGATED METHODSFOR SOLVING TWO-DIMENSIONAL INCOMPRESSIBLE FLOWS

EMPLOYING UNSTRUCTURED GRIDS

Hermínio Tasinafo Honório, [email protected] R. Maliska, [email protected] Fluid Dynamics Lab - SINMEC, Federal University of Santa Catarina - UFSC, Florianópolis/SC, Brasil

Abstract. In this work, a detailed comparison of the performance of different solution techniques to solve the momentum

and mass conservation equations for bidimensional incompressible flows is realized. This kind of comparison is common

in the literature, but, in this case, it was done using unstructured grids in the framework of the Element-based Finite

Volume Method. Unstructured grids, as very well known, requires the use of co-located arrangements of variables in

order to avoid excessive complexity of the code. The interpolation function of the FIELDS method was used to overcome

the checkerboard pressure problem, which arises from the co-located arrangement. Coupled solution, as well as the

solution involving pressure-velocity coupling methods where used and compared. When compared to the other methods,

the coupled solution has shown a great numerical stability and reduced computational effort most of the time.

Keywords: unstructured grids, segregated solution, coupled solution, element based finite volume method.

1. INTRODUCTION

Real engineering and technological problems, such as 3D, viscous and advective-dominated flows involving complexphysical phenomena like compressibility, shock waves and turbulence, started to be solved only after the second halfof XX century through the development of numerical techniques for solving partial differential equations. Particularly,for the Navier-Stokes set of equations, two numerical strategies are well known and mostly applied for its solution: thesegregated and the fully-coupled methods. Segregated methods remained as the best alternative for more than threedecades, since there was no computational facilities for using routinely coupled solvers.

Basically, segregated methods propose the solution of each conservation equations independently, obtaining the so-lution for one specific variable. As the iterative process goes on, all variables must gradually satisfy all conservationequations. For compressible flows this type of methods works pretty well since each variable has a clear evolutive equa-tion. That is, the velocities are calculated by the momentum equations, the density by the mass conservation equation andthe pressure by the equation of state. For incompressible flows, however, the equation of state is no longer suitable forthe pressure calculation, since it does not depend on density anymore. Additionally, pressure does not show up in massconservation equation. It gives rise to the pressure-velocity coupling problem the segregated methods have to deal with.Some of the most known methods of this type are the SIMPLE (Patankar and Spalding, 1972), SIMPLEC (Doormaal andRaithby, 1984), SIMPLER (Patankar, 1981) and PRIME (Maliska, 1981), which will be considered in this work.

On the other hand, if one decides to solve all conservation equations simultaneously in one system of algebraic equa-tions, necessarily, all variables must fully satisfy all conservation equations at the same time. Thus, there is no need of apressure-velocity coupling method. In this case, iterations are still needed only to take into account for the non-linearities.Macarthur and Patankar (1989) compared different techniques to handle the non-linearities and concluded the successivesubstitution (Picard method), despite its simplicity, is almost as efficient as the Newton and Broyden methods. Anotherproblem that shows up is how to handle the zeros in the main diagonal of the system of equations due to the absenceof pressure in continuity equation. A common way to overcome this problem is the employment of a proper interpola-tion function that considers the neighboring pressure effect on the velocities (Schneider and Raw, 1987; Rhie and Chow,1983). This technique is of special interest since it introduces pressure into the continuity equation and, at the same time,helps to avoid the checkerboard problem (Patankar, 1980).



The segregated methods give rise to smaller systems of equations when compared to the fully-coupled methods. Thus,specially for finer grids, it requires less computational memory and, for this reason, it has always been preferred. Thecomputational capabilities available nowadays, however, allow fully-coupled algorithms to be run without compromisingcomputational performance. The discussion about the performance of both type of methods is not sufficiently addressedin the literature, specially when unstructured co-located grids are employed. Thus, since it is an open subject for debate,this work aims to perform a systematic comparison between these two types of methods in order to contribute for fillingup this gap and to deep the understanding of their behavior.

2. NUMERICAL TECHNIQUES

Firstly proposed by Baliga (1978) and later improved by Schneider and Raw (1987), in the element based finite volumemethod (EbFVM) the control volumes (CV) are constructed around every grid nodes (element vertices) by the union ofall sub-control volumes (SCV) that share a same element vertex, as depicted in Figure 1. All sub-control volumes arebuilt based only on the information contained in the element they belong to. Also, the fluxes in the integration points (j),required by the conservation equations, can be evaluated based on the same element information. This allows an elementby element approach that, together with the element shape functions, is the key for handling unstructured grids.

Node4or4element4vertex4(i)

Integration4point4(j)

Element4(e)

Sub-control4volume4(SCV)

Control4volume4(CV)

SCV1

SCV2 SCV3

SCV4

Figure 1: Control volume construction and geometric entities.

The use of element shape functions by the EbFVM is a feature inherited from the finite element method. These shapefunctions transform an irregular element in the global coordinate system into a regular element, with fixed dimensions, inthe transformed coordinate system. In practice, they allow the interpolation of any quantity stored at the element verticesin any position inside the element. In general, quantities are needed to be interpolated only at the integration points ofthe faces and at the centroids of the sub-control volumes, where the local coordinates are known and, therefore, the shapefunctions can be calculated. For example, supposing Φi is a scalar stored at the vertices (i) of a given element, the valueφj evaluated at the integration point (j) of an element face is given by,

φj =

NEV∑i=1

NijΦi (1)

where Nv is the number of element vertices. The sub-index i of Nij refers to the shape function associated with the vertexi of the element and it is evaluated at the local coordinates of the integration point j.

The gradient of a scalar can also be obtained from the spatial derivatives of the shape functions, as shown in equation2. As it can be seen, all derivatives can be evaluated at any point inside the element based only on the values stored at itsvertices, which is indispensible for the dicretization processes.

~∇φj =

NEV∑i=1

[∂Nij

∂x

∂Nij

∂y

]TΦi (2)

Beyond being the main tools for obtaining the algebraic representation of the differential equations, as will be seen ahead,the Equations 1 and 2 allow some geometric calculations, such as the volumes Ω of the control volumes and the area



vectors ~S defined on faces. These calculations are shown by Schneider and Raw (1987).

2.1 Velocity interpolation function

It is well known (Maliska, 2004) that, when low Reynolds numbers are involved, the 2o order interpolation schemes,such as the central differencing scheme (CDS), are suitable since diffusion dominates the flow and, therefore, the down-stream values, as well as the upstream ones, affect the property value at the integration point. When the Reynolds numberbegins to rise, the advection terms come into play causing the 2o order schemes to show oscilations (and instabilities)near sharp gradient regions, due to their non-dissipative nature. In these cases (high Reynolds) 1o order schemes, suchas the upwind differencing scheme (UDS), are more suitable since they are stable and do not introduce oscilations in thesolution, although they do present the "false difusion" problem.

The above discussion leads to the idea of a hybrid method that gradually switches from 1o to 2o order as the Reynoldsnumber starts to decrease. In another words, an interpolation scheme that suitably balances the advection and diffusioneffects. However, as discussed by Raw (1985), in the case of momentum equations the inclusion of the pressure gradienteffects are of utmost importance for also avoiding the "false diffusion" problem. The interpolation function of the FIELDSmethod, developed by Schneider and Raw (1987), naturally takes these effects into account by using the differential formof the momentum equations, evaluated at the integration point, to its deduction. For the x-component of the velocityvector, u, for example, one may start by the following equation,

ρ|~Vj |∂u

∂s

∣∣∣∣j

= − ∂p

∂x

∣∣∣∣j

+ µ∇2u∣∣j

(3)

where ρ is the fluid density, µ is the fluid viscosity and s is the stream line direction at the integration point. Whenproperly discretized, Equation 3 will give the value of uj as a function of U and P (x-component of the velocity vector andpressure, respectively) stored at the element vetices, i, that contains the integration point, j. Performing the discretization,the following expression is obtained,

uj =

NEV∑i=1

(γij

ψjUi −

1

ψj

∂Nij

∂xPi

)(4)

The Equation 4 gives an expression of uj as a function of Ui and Pi weighted by the shape function derivatives andthe coefficients ψj and γij. These coefficients take advection and diffusion effects into account and their expressions anddeduction can be seen in Honório (2013).

3. MATHEMATICAL MODEL AND NUMERICAL FORMULATION

In this work, as already mentioned, different solution strategies were employed for the solution of two-dimensionalincompressible flows. Also considering single-phase flow, no heat transfer or chemical reactions, newtonian fluid and con-stant viscosity, the mathematical model is comprised only by the continuity equation (5) and the momentum conservationequations (6) as the following,

~∇.~V = 0 (5)

∂

∂t

(ρ~V)

+ ~∇.(ρ~V ~V T

)= −~∇.P + ~∇.

(µ~∇~V T

)(6)

where P = pI . For cartesian components, Equation (6) results in both x and y momentum equations.The discretization procedure for obtaining the algebraic equations starts by a time integration followed by a volume

integration of Equations (5) and (6). The total implicit formulation (Maliska, 2004) were employed to evaluate the time



integrals. The divergence theorem is applied to transform the control volume integrals into surface integrals. The surfaceintegrals are taken by parts at each face of the controle volumes. The integrands of the surface integrals are assumed to beconstant over each face, so they can be evaluated at the center of the faces (integration points). This procedure leads to thefollowing discretized representation of Equations 5 and 6 for a control volume centered at a node p (or element vertex),

NFP∑j=1

(Sxj uj + Sy

j vj)

= 0 (7)

MpUp

∆t+

NFP∑j=1

mjuj =

NFP∑j=1

NEV∑i=1

(µβijUi − Sx

jNijPi

)+Mo

pUop

∆t(8)

MpVp∆t

+

NFP∑j=1

mjvj =

NFP∑j=1

NEV∑i=1

(µβijVi − Sy

jNijPi

)+Mo

pVop

∆t(9)

where NFP is the number of faces surrounding the control volume p, Mp is the control volume mass, mj is the mass fluxcrossing the face j and the superscript o denotes the evaluation of a property at the previous time step (∆t). The x and ycomponents of the area vector are being represented by Sx

j and Syj . The diffusive term of the momentum equations (last

term of the right-hand side of Equation 6) were evaluated using Equation 2, which originated the diffusive operator βij,given by,

βij =∂Nij

∂xSxj +

∂Nij

∂ySyj (10)

Equations 8 and 9 also show that the shape functions Nij were used to represent the pressure value at an integration pointj as a function of its nodal values Pi. What is left now is to adequately represent uj and vj , which is done by the velocityinterpolation function of the FIELDS method, discussed in the last section. Then, by substitution of the Equation 4 foruj and a similar one for vj into the Equations 7, 8 and 9 and conveniently grouping the terms, the dicretized forms of thegoverning equations are finally obtained,

NFP∑j=1

NEV∑i=1

(App

ij Pi +Apuij Ui +Apv

ij Vi)

= 0 (11)

MpUp

∆t+

NFP∑j=1

NEV∑i=1

(Auu

ij Ui +Aupij Pi

)=Mo

pUop

∆t(12)

MpVp∆t

+

NFP∑j=1

NEV∑i=1

(Avv

ij Vi +Avpij Pi

)=Mo

pVop

∆t(13)

More details on the deduction of the above equations can be found in (Honório, 2013).

3.1 Solution Strategies

As stressed out by Maliska (2004), solving Equations 11 to 13 by a segregated method means to find a pressure field(P ) that when introduced into the momentum equations (12 and 13) results in a velocity field (U and V ) that satisfies notjust the momentum equations, but also the continuity equation (11). In fact, this will happen only when the pressure field



(P − V coupling) and the coefficients (non-linearities) of the momentum equations are correct. There are a number ofstrategies for achieving that and each one of them deals with these two points in different manners.

The PRIME method, for example, solves the Equation 11 implicitly for pressure (using guessed U and V ). Thispressure field is then introduced into the momentum equations, which are solved in a explicit manner to obtain a newvelocity field. This new velocity field satisfy only partially the momentum equations and it will satisfy the continuityequation only when the iterative process is converged. In this method, the only mechanism that "guides" the velocity fieldto satisfy the continuity equation is the pressure field introduced into the momentum equation, since it is obtained fromthe mass conservation equation. For this reason, although there is just one linear system being solved at each iterationstep, it is expected a slow convergence rate.

The SIMPLE-like methods (SIMPLE, SIMPLEC and SIMPLER), on the other hand, ensures that the velocity fieldsatisfies the continuity equation at the end of every iteration cycle. In these methods, the momentum equations (12 and 13)are solved implicitly with a guessed pressure field to obtain a velocity field. This velocity field, which satisfy momentumequations, must then be corrected in order to satisfy the continuity equation. So, at one iteration cycle, the velocity fieldsatisfies the momentum equations and then the continuity equation, but not both at the same time, except when the iterativeprocess is converged. The equations for correcting the velocities are based on the definition of a pressure correction field,defined as P ′ = P − P ∗, where P ∗ is the pressure field of the previous iteration step. An equation for obtaining thepressure correction field can be obtained through the continuity equation (7) using correction equations, obtained with theaid of the FIELDS interpolation functions, to evaluate the integration point velocities (uj and vj). This leads to,

NFP∑j=1

NEV∑i=1

Ap′p′

ij P ′i

=

NFP∑j=1

NEV∑i=1

[Ap′u

ij U∗i +Ap′uij V ∗i −A

p′pij P ∗i

](14)

which is also solved implicitly. Once the pressure correction field is obtained, the velocities stored at the element verticesare also corrected to ensure the mass conservation in each control volume. The equation used to correct the U velocity,for example, is

Up = U∗p − dupNFP∑j=1

NEV∑i=1

(Aup

ij P′i

)(15)

where dup = 1/Aup for SIMPLE and SIMPLER methods, and dup = 1/

(Au

p +∑

nb Aunb

)for the SIMPLEC method. A

similar equation is also used to correct the V velocity. The velocity correction equations are the way the SIMPLE-likemethods treats the non-linearities of the momentum equations, which is not done by the PRIME method. The completededuction of Equations 14 and 15, as well as their coefficients, can be found in Honório (2013).

Once the linear systems of Equations 12 and 13 are solved, the pressure correction field is obtained from Equation14 and the velocities at the element nodes are corrected, a new pressure field must be calculated to be used in the nextiteration step. With this purpose, the SIMPLER method solves another linear system, originated from equation 11,while the SIMPLE and SIMPLEC methods explicitly calculates a new pressure field using the pressure correction fielddefinition,

P = P ∗ + αP ′ (16)

where α is a sub-relaxation factor. As suggested by I. Demirdzic and Peric (1987), in this work, α was set to 1/ (1 + E),where E is proportional to the time step, as explained by Doormaal and Raithby (1984)). It is important to stress that theSIMPLER, SIMPLE, SIMPLEC and PRIME methods solve, respectively, four, three, three and one linear system at eachiteration cycle. Each linear system has dimensions NCV ×NCV, being NCV the number of control volumes of the grid.

The other strategy considered in this work was the coupled solution technique. In this method, the three Equations11, 12 and 13 are solved altogether in just one system of equations of dimensions 3NCV × 3NCV, which has the structure



presented in Equation 17. Because all the variables are solved implicitly, the velocity field obtained fully satisfy thecontinuity and the momentum equations so it is tightly coupled with the pressure field. However, the velocities used tocalculate the coefficients are wrong due to the non-linearities issue. Therefore, the system of equations 17 has to be solvediteratively until the convergence criteria is reached.

Continuity

Momentum− xMomentum− y

−→−→−→

App Apu Apv

Aup Auu 0

Avp 0 Avv

PUV

=

0

Bu

Bv

(17)

4. RESULTS AND DISCUSSION

Once the algorithms implemented were validated, tests were carried in order to assess the performance of both fully-coupled and segregated techniques. The efficiency of a method can be understood as directly proportional to stability androbustness, and inversely proportional to the computational effort (CPU time consumed). For all methods, stability andCPU time are dependent of the time step chosen (or the E parameter) since it acts as a relaxation parameter to reach thesteady state solution. Besides, the methods studied deal with different numbers of system of equations, with differentsizes. So it is expected the increasing of CPU time to be different for each method as the grid is refined, that is, methods’performances may be grid dependent. One of the problems considered in order to test these different scenarios was thewell known lid driven cavity problem for Re = 100. As it can be seen in Figure 2a, the fully-coupled and the SIMPLEmethods showed a good stability compared to the other methods since they converge for a wider range of the E parameter.As the grid is refined, Figure 2b showed a great performance of the fully-coupled technique followed, again, by theSIMPLE method. Based on both graphics of Figure 2, it can be said there is no advantage in solving a laplacian equationfor pressure as it is done by the PRIME and SIMPLER methods.

0 5 10 15 20 250

500

1000

1500

2000

2500

3000

3500 PRIME

SIMPLE

SIMPLEC

SIMPLER

Coupled

CP

U T

ime

(s)

E parameter

(a)

103100

101

102

103Coupled

PRIME

SIMPLE

SIMPLEC

SIMPLER

n)-)Number)of)nodes

CP

U)T

ime)

(s)

(b)Figure 2: These figures show the CPU time behavior of all methods with respect to (a) the E parameter (which is propor-tional to time step) and (b) the grid refinement.

Also, the methods’ robustness were analyzed through the same problem with Re = 3200, which is a highly advective-dominated flow. Again, Figure 3 shows expressive oscillations in pressure residue of both PRIME and SIMPLER, whichshould not happen since they solve a system of equations exclusively for pressure. At this time, the SIMPLE method tookless iterations to reach the convergence criteria (which is when the euclidean norm of |U − Uo| and |V − V o| reaches avalue below 10−4) than the fully-coupled one. The most important fact observed was the SIMPLEC’s lack of stability androbustness, as shown by Figures 2a and 3. It was found that this method never converge when in momentum equationsAp +

∑nb Anb < 0 in at least one control volume. This causes a bad conditioning in the pressure correction system of

equations that prevents its solution. It can be overcome by choosing a small time step in a way that Ap > −∑

nb Anb.However, it considerably slows down the convergence rate, as it can be expected when small time steps are employed.This is the cause of the slow convergence observed in Figure 3 for SIMPLEC. The detailed reasons for this to happen will



be addressed soon in another paper.

Pre

ssu

re R

esid

ue

IterationsFigure 3: Pressure residue reduction for the lid driven cavity problem with Re = 3200.

The bidimensional flow inside a ball valve was also solved. The grid employed and the boundary conditions imposedare shown in Figure 4. Despite not having a reference solution, this problem is of interest because it represents a casewhere there is mass flux in and out of the domain and it has a complex flow pattern, with some regions of recirculation.Also, the irregular geometry illustrates a domain that would be difficult to be discretized using structured grids.

Figure 4: Pressure residue reduction for all methods as the iterative process goes on.

This problem was also solved by all the methods presented before. The convergence criteria was the same for the liddriven cavity. Figure 5 shows the pressure residue drop for each method. As expected, the convergence rate of the PRIMEmethods was lower than the other methods, except for SIMPLE. At this time, the SIMPLE method was very unstableand required a small time step (causing the slow convergence rate) to reach the convergence criteria. As it can be seen,the SIMPLEC and SIMPLER methods took the same number of iterations and showed the best performances among thesegregated methods. The coupled solution took only 9 iterations to reach the convergence criteria, showing excellentrobustness and stability.

Pre

ssu

re R

esid

ue

IterationsFigure 5: Pressure residue reduction for all methods as the iterative process goes on.

5. CONCLUSIONS

The most known segregated techniques and the fully-coupled solution were tested and compared on the element basedfinite volume method (EbFVM) framework. For the mathematical model considered, it was found that taking the pressure-velocity coupling implicitly in the system of equations, as it is done by the fully-coupled technique, brings great robustnessand stability for the iterative process. Despite having a system of equations three times bigger, the coupled solution isusually less time consuming than the segregated methods and, therefore, more efficient. The SIMPLE method was the



most efficient among the segregated methods, except for the ball valve problem. It was not expected the instabilitiespresented by the SIMPLEC method. These instabilities are caused by the way the mass fluxes are calculated at the controlvolume faces, but only the SIMPLEC method was affected by that. Both methods that solved a linear system to obtainthe pressure field (PRIME and SIMPLER) did not show a good performance most of the time. In general, it could beobserved that the segregated methods’ efficiency has a certain dependency on the problem to be solved. In this sense, thefully-coupled technique is more consistent and reliable than the segregated ones, since its performance did not decreasedrastically in any situation.

6. ACKNOWLEDGEMENTS

This work was supported by Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES).

7. REFERENCES

Baliga, B.R., 1978. A Control-Volume Based Finite Element Method for Convective Heat and Mass Transfer. Ph.D. thesis,University of Minnesota.

Doormaal, J.P.V. and Raithby, G.D., 1984. “Enhancements of the simple method for predicting incompressible fluidflows”. Numerical Heat Transfer, Vol. 7, pp. 147–163.

Honório, H.T., 2013. Análise de Métodos Segregados e Acoplado de Solução de Escoamentos Incompressíveis Utilizando

Malhas Não-Estruturadas Híbridas. Master’s thesis, Universidade Federal de Santa Catarina.I. Demirdzic, A. D. Gosman, R.I.I. and Peric, M., 1987. “A calculation procedure for turbulent flow in complex geome-

tries”. Computer and Fluids, Vol. 15, pp. 251–273.Macarthur, J.W. and Patankar, S.V., 1989. “Robust semidirect finite difference methods for solving the navier-stokes and

energy equations”. International Journal for Numerical Methods in Fluids, Vol. 9, pp. 325–340.Maliska, C.R., 1981. A Solution Method for Three-Dimensional Parabolic Fluid Flow Problem in Non-orthogonal Coor-

dinates. Ph.D. thesis, University of Waterloo.Maliska, C.R., 2004. Transferência de Calor e Mecânica dos Fluidos Computacional. LTC - Livros Técnicos e Científi-

cos, Rio de Janeiro, 2nd edition.Patankar, S.V., 1981. “A calculation procedure for two-dimensional elliptic situations”. Numerical Heat Transfer, Vol. 4,

pp. 409–425.Patankar, S.V. and Spalding, D.B., 1972. “A calculation procedure for heat, mass and momentum transfer in three-

dimensional parabolic flows”. Int. J. Heat Mass Transfer, Vol. 15, pp. 1787–1806.Patankar, S.V., 1980. Numerical Heat Transfer And Fluid Flow. Hemisphere Publishing Corporation.Raw, M.J., 1985. A New Control Volume-Based Finite Element Procedure for the Numerical Solution of the Fluid Flow

and Scalar Transport Equations. Ph.D. thesis, University of Waterloo.Rhie, C.M. and Chow, W.L., 1983. “Numerical study of the turbulent flow past an airfoil with trailing edge separation”.

AIAA Journal, Vol. 21, pp. 1525–1532.Schneider, G.E. and Raw, M.J., 1987. “Control volume finite-element method fot heat transfer and fluid flow using

colocated variables - 1. computacional procedure”. Numerical Heat Transfer, Vol. 11, pp. 363–390.

8. RESPONSABILITY NOTICE

The authors are the only responsible for the printed material included in this paper.

on the performance of coupled and segregated …€¦ · scv1 scv2 scv3 scv4 figure 1: control...

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